A comparison of pitch measurements on an oscillation towed body using a vertical gyro and a pendulum indicator

Research and Development Report

July 1957

_{Report 1153} Lab.

v.

Scheepsbouwkunde7

Technische Hogeschool

Delft

NAVY DEPARTMENT

THE DAVID W. TAYLOR MODEL BASIN

WASHINGTON 7. D.C.

A COMPARISON OF PITCH MEASUREMENTS _{ON AN OSCILLATING}

TOWED BODY USING A VERTICAL GYRO _{AND A} PENDULUM INDICATOR

by

A COMPARISON OF PITCH MEASUREMENTS ON AN OSCILLATING TOWED BODY USING A VERTICAL GYRO AND A

PENDULUM INDICATOR by

Samuel M. Y. Lum and Chester 0. Walton

TABLE OF CONTENTS Page ABSTRACT INTRODUCTION 1 EXPERIMENTAL SET-UP DISCUSSION OF RESULTS BASIN TESTS DYNAMIC CALIBRATION

CONCLUSIONS AND RECOMMENDATIONS

ACKNOWLEDGMENTS 8

APPENDIX A. - THE DAVID TAYLOR MODEL BASIN HEAVING FACILITY

APPENDIX B. - ANALYSIS OF THE EFFECTS OF TOWED BODY MOTION 11

ON A PENDULUM ANGLE INDICATOR

4

7

Figure 4b Figure 4e Figure 4d Figure

5

-Figure 6a Figure 6b Figure 6c Figure 6d Figure 7 -LIST OF ILLUSTRATIONS

Figure I - Towed Body Used in Experiment Figure 2 - The Heaving Towpoint Facility

Figure

3 -

Experimental Set-Up Showing Dynamic Calibration of

Gyro and Pendulum Indicator

Figure 4a - Sample Records Showing Pitch Response of Towed Body as Indicated by Gyro and by Edcliff Pendulum, w = 0

Radian per Second

Sample Records Showing Pitch Response of Towed Body as Indicated by Gyro and by Edcliff Pendulum, w = 2.0

Radians per Second

Sample Records Showing Pitch Response of Towed Body as Indicated by Gyro and by Edcliff Pendulum, w - 2.4

Radians per Second

Sample Records Showing Pitch Response of Towed Body as Indicated by Gyro and by Edcliff Pendulum, m = 2.8

Radians per Second

Comparison of Maximum Pitch Amplitudes of Towed Body as Measured by Gyro and Pendulum for Different Input

Frequencies

Sanborn Traces of Gyro and Pendulum Dynamic Calibration,

w = 2.02, 2.17 Radians per Second

Sanborn Traces of Gyro and Pendulum Dynamic Calibration,

w = 2.26, 2.38 Radians per Second

Sanborn Traces of Gyro and Pendulum Dynamic Calibration,

w = 2.49, 2.57 Radians per Second

Sanborn Traces of Gyro and Pendulum Dynamic Calibration,

U) = 2.81,

2.88

Radians per Second

Effect of Horizontal Acceleration and Frequency on Gyro

and Pendulum Output

-A COMP-ARISON OF PITCH ME-ASUREMENTS ON -AN OSCILL-ATING TOWED BODY USING A VERTICAL GYRO AND A

PENDULUM INDICATOR by

Samuel M. Y. turn and Chester 0. Walton

ABSTRACT

A comparison is made of the relative accuracy of a pendulum indicator as compared with a vertical gyro in determining the

pitching motion of a cable-towed body under forced oscillation.

It is shown that serious errors are introduced in the measurement of body attitude with a pendulum indicator when the body is

sub-jected to accelerating forces.

INTRODUCTION

In the determination of the pitching response of a cable-towed body subjected to an oscillating input, pendulum indicators

are frequently used to measure angular orientation of the body.

Generally, space availability, the number of conductors required, simplicity of operation and recording, and cost economy have

dictated the use of pendulums. Since, however, these devices

were designed only for static measurements, the question of their

accurac under dynamic conditions has been cause for serious

con-cern. Recently, during the course of obtaining motion data on a

cable-towed body, the Model Basin had occasion to compare a

pendulum indicator with a vertical gyro under dynamic conditions. This report presents the results of this comparison.

EXPERIMENTAL SET-UP

The specifications for the gyro and pendulum units are given

in Table 1. The towed-body in which they were installed is shown in Figure 1. This body was constructed for another purpose and

was used for these tests merely as a matter of convenience. As

a result, no particulars need be given except that the body was

21

feet long and contained internal ballast to provide metacentric

stability and permit trimming to zero in the submerged condition. The gyro and pendulum were located as shown in the following sketch.

The tow cable besides being a strength member provided sufficient electrical conductors for powering the gyro and

pendulum and for transmitting the output signals to a 2-channel

pen recorder. The cable was connected to the TMB heaving towpoint facility shown in Figure 2 and described in detail in Appendix

A.

Basin tests consisted of simultaneously recording the out-puts of the gyro and pendulum with the body towed at various

speeds and subjected to an oscillating input at the towpoint.

The vertical motion of the towpoint was varied over a frequency range of 0 to 2.8 radians per second with a constant amplitude of

1 foot.

In addition to basin tests, it was deemed advisable to also

make a dynamic calibration of the gyro and pendulum units. This

was accomplished by means of the swing shown in Figure 3. Both

indicators were located in the box, as shown, and subjected to known motions controlled by the variable speed motor and measured

accurately with the angular potentiometer. Comparison was made

between the angle measured by the potentiometer and that indicated

Instrument Manufacturer Type Cost (Approx.) Power Accuracy (stati Principle Pendulum Edcliff Duarte, California $160 DC Battery TABLE 1

Comparison of Gyro and Pendulum

0.5 deg

Pendulum depends on gravity to give the vertical reference. The pendulum mass supported by instru-ment type ball bearing, actuates a potentio-meter producing an output linear with rotation. Liquid in case provides fluid damping. 3 Gyro Minneapolis-Honeywell Minneapolis, Minnesota Vertical, Self-Erecting JG-10414A $1,085 AC-DC 115v.,

400

cps single phase 4 0.15 deg Rapid spinning of gyro rotor provides stable reference. Long term drift com-pensated by mercury switches which actu-ates erection motors to maintain vertical alignment of gyro spin axis. Conductors Req'd I 3 Model Ident. 5-5-1 5 1/8" x 5

7/8" x 7 3/4"

5.0 lb Size 2.0" dia. x 1.1" Weight 0.75 lb

BASIN TESTS

The results of tests with the towed body are given in Table 2.

The sign convention is plus for nose-up attitude and minus for

nose-down.

TABLE 2

Comparison of Recorded Body Pitch from Gyro and

Pendulum Indicator

DISCUSSION OF RESULTS

Carriage Speed in Knots

0

+.2

1+4.2,- 4.4

-6.0,+ 8.6

L _

+3.80- 6.2

+5.2,- 5.0

-(.3,+23.0

Figures 4a thru Lld show sample records of the simultaneous outputs from the pendulum and gyro for the various test conditions.

Analysis of these records shows that the pendulum output is quite

irregular,

particularly

at the higher speeds. Furthermore, the

pitch

recorded by the pendulum is not in phase

Irith

the gyro output.

At 10 knots and the higher frequencies this phase shift

iP about

180 degrees.

Figure 5 shows 9 compavison of the absolute values of

max-imum pitch amE:itude

(10mald)

measured by the gyro and ,Jer!':,11:m for various input frequencies (w). At low speeds and frequencies the pitch measured by the pendulum is within 2 or 3 degrees of tht

measured by the gyro. At the higher speeds and frequencies, howe,'e,,

Input Frequency in radians per second V1 = 2.5 1

V2 = 5.0

v3 - 7.5

co

_l

=

Gyro

Pend

+2.0

+1.5

+1.0

a)2

= 2.0

Gyro +7.0,-11.2 Pend +7.2,-

9.8

+6.6,-

(.2

+6.o,- 8.0

+5.0,- 6.0

4-4.2,- 7.0

w, = 2.4

Gyro

+8.4,- 8.2

Pend

+8.4,- 7.3

+6.0,- 9.2 +5.4,-10.1 +5.0,- 6.1

+8.o,- 7.0

a

- 2.8

Gyro

+13.2,-14.8

Pend

+13.0,-13.0

+11.0,-10.G+7.5,-10-O

+7.o,- 6.8

-8.2,+15.8

-.3

0

the pendulum shows a pitch amplitude as much as 4.5 times higher

than that of the gyro. It should also be noted that, in the plot

of 10,,,x1 versus w, the variation of pitch response with frequency

shown by the pendulum is entirely different from that shown by the

gyro.

DYNAMIC CALIBRATION

The results of the dynamic calibration performed under known angular inputs in the laboratory are given in Figure 6a thru 6d

inclusive. These records show that the gyro gives an accurate measure of pitch both in amplitude and phase. All gyro readings

were within ±i-degree of that of the angular potentiometer with apparently no phase difference whereas the pendulum record shows amplitude errors of as much as 17 degrees and phase differences as

great as 180 degrees.

To gain some insight into the effect of acceleration on the gyro and pendulum, the horizontal component was computed for the

conditions of dynamic calibration. In this case, the horizontal

acceleration is due entirely to the angular motion of the box in

which the gyro and pendulum were mounted. In the following sketch (See Figure 3 also)

P is the position of the transducer, P(i,)

is the length of the swinging arm, 5P

e is the angular displacement of the arm at any time t, is the horizontal displacement of P at any time t,

and co is the circular frequency of the oscillating input.

Taking fixed axes and C as shown, the horizontal displacement of the point P with respect to the origin at 0 is given by

= 2 sin e

The second derivative with respect to time is then

d2 = (g cos 8 - 62 sin (9) _[1]

dt2

For a harmonic input oscillation, the angular displacement is given;

by

= 00 sin cot

where

ea

is the maximum angular travel of the swinging arm.

Substituting; the horizontal acceleration of the point P

is given by:

= -2060 [sin cot cos (9 + 60 cos2wt sin (3]

(2n-1)T

The maximum acceleration occurs at

e = eo

or at time t

4

where T is the period and n = 1, 2,

3,

- - -

Also, 27

the circular frequency by definition is w

= T.

Substituting, we

obtain for the absolute value of the maximum horizontal acceleration:

6

-Imaxl = 1.00 w2

I"41axl = Lw29 cos

In this particular case, = 6 ft. and ao = 9.7 degrees - .169 radians whence:

win rad/sec

k in ft/sec2

From this relationship, the values of gmaxl were computed for each

emax

frequency w and plotted against values of for the gyro and Go

pendulum.

This comparison is given in Figure 7. As can be seen from the

sample calibration records given in Figure 6a thru Od, it was

difficult to determine a value of 9ma, for the pendulum. The value selected for each w was taken from an average of the recorded peaks.

The significance of such a value is, however, subject to considerable

question. Consequently, Figure 7 serves only to qualitatively il-lustrate the effect of frequency and horizontal acceleration on the pendulum error. It also clearly shows that there is no such effect

on the gyro over the range of values considered.

To further illustrate the effect of body acceleration on the

accuracy of pendulum indicators, an analysis was made for the case

of the towed body subjected to motions in the vertical plane. This analysis is presented in Appendix B.

CONCLUSIONS AND RECOMMENDATIONS

The results of this comparison between a gyro and pendulum indicator clearly demonstrate the danger in using pendulums to

measure angles under dynamic conditions. The range of input motions

applied to the towed vehicle used in making this comparison is typical of motions which could occur in towing such bodies from a

vessel operating in a seaway. This appears to preclude the use of pendulums for other than static measurement of angular attitude.

It is recommended, therefore, that future evaluations of cable-towed bodies such as the variable-depth-sonar be based on the use of

ACKNOWLEDGMENTS

The authors are indebted to Messrs. M. Graybill, J. Leahy and E. Frillman of the Applied Instrumentation Branch for their generous assistance in providing the instrumentation necessary

to make the measurements described in this report.

APPENDIX A. - THE DAVID TAYLOR MODEL BASIN

HEAVING TOWPOINT FACILITY

General Description and Purpose

The "heaving towpoint" is a device by means of which systematically varied vertical displacements can be applied

to the end of a cable from which a submerged body is towed.

This device is design to be attached to TMB Carriage 2 near

the 4, of the basin. This carriage has a top speed of about

17 knots. The purpose of this towpoint is to provide a means

of simulating the vertical motions of a real towing platform

such as a ship. The stability characteristics of towed bodies such as the variable depth sonar can thus be predicted more accurately from model tests.

Design Specifications

The tow point oscillates along an axis which is normal to a plane tangent to the earth's surface,

i.e., a line usually referred to as the vertical axis The total vertical displacement can be varied from 0

to 10 feet and the center of oscillation can be positioned

anywhere along the 10-foot length. It should be noted,

however, that the maximum amplitude is realized only when the center of oscillation is in the center of

the 10-foot length.

Either a single step function or a continuous

sinusoidal motion can be produced.

For the sinusoidal motion the double amplitude is variable from 0 to 10 feet and the period is

variable from about 3 seconds to infinity.

The maximum design loading on the towpoint is

1000 pounds horizontal (drag) force, 1000 pounds

vertical force (including dead load) and 500

pounds side force.

Maximum Permissible Model Size

The maximum size of model to be used on this facility will

be determined by the maximum loads specified above. These

loads are a function of several factors; namely, the shape of the body, the length of cable, the mass of the body, the accelerations which are to be imposed, and the maximum towing

speed. It is seen, therefore, that the maximum model size can

We can, however, obtain some idea of the limits imposed on the

simulation

of

full-scale parameters. This can be

done

since

we know that the laws of dynamic scaling will be followed in

determining full-scale values based on model tests. In other

words, when we model a dynamic towing test we preserve the

full-scale value of the ratio V__, where V is the speed of

igL

advance, L is a characteristic length dimension, and g is the

acceleration

of

gravity. On this basis, the

following

scale relations are obtained:

For a model scale

of X,

Full-scale length X -Model-scale length Full-scale time Model time -x2 Full-scale velocity Model velocity x2

Model linear accelerations = Full-scale linear accelerations

Model angular accelerations - Full-scale angular accelerationsq

Full-scale forces Model forces

-X3

As a result, the maximum full-scale loads which can be duplicated

are:

Full-scale horizontal (drag) force = 1000 X3 pounds

Full-scale vertical force = 1000 X3 pounds

Full-scale side force = 500 X3 pounds

Within the allowable loads then, the towpoint can produce a

sinusoidal motion

with a maximum full-scale doublq-amplitude

of 10

X feet and minimum full-scale period of 3 A.-2 seconds,

with linear accelerations equal for both model and full-scale. 1

APPENDIX B - ANALYSIS OF THE EFFECTS OF TOWED BODY MOTION ON A PENDULUM ANGLE INDICATOR

In the comparison of the gyro and pendulum for the laboratory

calibration, the origin 0 was fixed in space. As a result, the

acceleration of the point P was due only to rotation about a fixed

point.

In order to interpret the results of the gyro and the pendulum mounted in the body when undergoing oscillations under the sinusoidal

input from the heaving towpoint, it is necessary to consider an additional horizontal acceleration component at the point P in

addition to that at the origin,O. Here, the point 0 is in a movIng

axes system with freedom in translation as well as rotation. Using axes fixed to the body, we take Y positive toward the nose of the body and z positive downwards perpendicular to x, as in the follnwL1g

sketch:

We may define:

b = distance of point P from origin 0

e and C coordinate axes fixed in space

The instantaneous velocity of the origin may then be written

= lu + 'RIAT

where u and w are the velocity components along the respective x

and z body axes assuming two dimensional motion in the vertical

plane. The acceleration vector at 0 may be obtained by taking

the substantial derivative of the velocity vector,

D

yo = do

(E) x Vo)

Dt dt

where the vector cross product of the angular velocity vector aS

and the linear velocity vector Vo is given by

assuming only rotational motion about the transverse axis.

The acceleration of the towpoint with q = 0 can then be written as

D

vo = I

+ ew) + Tc (W - eu)

Dt

This must be combined vectorially with the linear component due to the angular acceleration of the point P rotating with respect to the origin 0 to get the total acceleration that the

transducer experiences.

The point P where the transducer is located can be referenced

to the origin by [2] 12

7

FD- x -70 0 q 0 u 0 w

-xp = b sin 0 = b sin (5-0)

Zp = b cos p = b cos (b-e)

The velocity and acceleration of point P which is fixed to

the body relative to the origin 0 can be obtained by taking the

first and second derivatives with respect to time.

= - IDE5

COS (5-e)

ip = be sin (5-9)

Kp = - bg cos (5-0) - bP sin (5-9)

2P -- be sin (5-e) - be2 cos (5-e)

[3c]

Combining the corresponding components from Equations [2] and

[3c], the acceleration of the point P may be given by its

longi-tudinal and vertical components along body axes as

ax 1.1 + we

-

be2 sin (s-(9) - bg cos (54)

[4]

az = *t;T

- ue -

be2 cos (5-e) + b.() sin (5-e)

Since the pendulum transducer works on the gravity principle,

we must refer the accelerations of the point P measured along body

axes to that of space axes. Using and C for the space axes as

shown in the preceding sketch, the relationship between the

accelerations along body and space axes are given by

= ax cos e + az sin

[5]

C = az cos 6 - ax sin

Considering only the horizontal space acceleration, the

resolution of can be obtained by substituting ax and az from

Equation [4] respectively in Eluation [5]. The resultant equation

for the horizontal space acceleration of the point P can be stated

as

-= U

This can be demonstrated by sliding the pendulum transducer

back and forth on a flat table to simulate pure surging. By such

means the mass in the pendulum could easily be made to move giving

erroneous readings.

However, in finally discussing the sum error of the pendulum transducer, the differential equations of motion of the mass in the transducer must be investigated assuming knowledge of the inertia, damping and spring constants in addition to the external

forces induced by the kinematics of the point P. The frequency

response of this subsidiary system of pendulum mass with fluid damping and friction can then be studied to show the errors in

magnitude and phasing of the output. The discussions previously mentioned are restricted to the kinematics of a point P where the

transducer is located to show the nature of the input to this

subsidiary compound pendulum system.

= U COS 0 * sin 9 + 9 (w cos - u sin 9)

- 13E52 sin

(e+p) - h5

cos

(e+p)

[6]

where (9113) = 5

To determine the magnitude of the horizontal space

accelera-tion, therefore, it is necessary to obtain u and w. Unfortunately, these were not obtained for the towing tests previously described.

By comparing Equations

[6]

to [1], however, it can readily be seen

that now there are more components in the expression for the

horizontal space acceleration. There is possibility of a greater

or less error in pitch measurement using the pendulum when there

is surging and heaving of the body. For the range of conditions

tested, the pendulum gave a pitch reading about 4.5 times that of

the gyro reading in the worst case. This can be attributed to the u component arising from increased surging at the higher speeds.

It is also evident from Eluation

[6]

that the location of the transducer characterized by the arm b and the angle p has an

influence on the magnitude of the acceleration. If the transducer

were to be located on the towpoint itself such that b = 0, the

latter two terms in [6] would drop out.

In addition, if there was no pitching of the fish for the

zero trim condition such that 9, 8, and 0 were zero, the horizontal space acceleration in Equation

[6]

reduces to one of pure surge,

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of

Towed Body as Indicated

by Gyro and by Edcliff Pendulum,

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Figure 5

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as Measured by Gyro and Pendulum for Different Input

Frequencies

3.0

Legend:

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